Question

Given an experimental design with four factors A, B, C and D where each factor has two levels. Derive the formula for effect estimation of main effect of ?, interaction effect of ??, ??? and ????. (please use (1),a,b,c,d,ab,ac,ad,bc,bd,cd,abc,abd,acd,bcd,abcd to represent the response)

Answer 1

Answer:

Given that,

Given an experimental design with four factors A, B, C and D where each factor has two levels.

Derive the formula for effect estimation of the main effect of A, interaction effect of AB, BCD and ABCD.

(please u(1),a,b,c,d,ab,ac,ad,bc,bd,cd,abc,abd,acd,bcd,abcd to represent the response)

Design matrix for experiment,

Run	Response	A	B	AB	C	AC	BC	ABC	D	AD	BD	ABD	CD	BCD	ACD	ABCD
1	(1)	-	-	+	-	+	+	-	-	+	+	-	+	-	-	+
2	a	+	-	-	-	-	+	+	-	-	+	+	+	-	+	-
3	b	-	+	-	-	+	+	+	-	+	-	+	+	+	-	-
4	ab	+	+	+	-	-	-	-	-	-	-	-	+	+	+	+
5	c	-	-	+	+	+	+	+	-	+	+	-	-	+	+	-
6	ac	+	-	-	+	-	-	-	-	-	+	+	-	+	-	+
7	bc	-	+	-	+	+	+	-	-	+	-	+	-	-	+	+
8	abc	+	+	+	+	-	+	+	-	-	-	-	-	-	-	-
9	d	-	-	+	-	+	+	-	+	-	-	+	-	+	+	-
10	ad	+	-	-	-	-	+	+	+	+	-	-	-	+	-	+
11	bd	-	+	-	-	+	-	+	+	-	+	-	-	-	+	-
12	abd	+	+	+	-	-	-	-	+	+	+	+	-	-	-	+
13	cd	-	-	+	+	+	-	+	+	-	-	+	+	-	-	+
14	acd	+	-	-	+	-	-	-	+	+	-	-	+	-	+	-
15	bcd	-	+	-	+	+	+	-	+	-	+	-	+	+	-	-
16	abcd	+	+	+	+	-	+	+	+	+	+	+	+	+	+	+

Effect=Constrat/(x. )

Where,

x=number of observations at each factor level

k=no.of factors=4

:

= [ a+ab+ac+abc+ad+abd+acd+abcd]-[(1)+b+c+bc+d+bd+cd+bcd]

:

=[(1)+ab+c+abc+d+abd+cd+abcd]-[a+b+ac+bc+ad+bd+acd+bcd]

:

[b+ab+c+ac+d+ad+bcd+abcd]-[(1)+a+bc+abc+bd+abd+cd+acd]

:

[(1)+ab+ac+bc+ad+bd+cd+abcd]-[a+b+c+abc+d+abd+acd+bcd]